Question

A monopolist sells its product in two isolated markets with demand functions

            P1 = 32 − Q1 and P2 = 40 − 2Q2

The total cost function is TC = 4(Q1 + Q2).

(a) Show that the profit function is given by

       π = 28Q1 + 36Q2 − Q12 − 2Q22

(b) Find the values of Q1 and Q2 which maximise profit and calculate the value of the

maximum profit. Verify that the second-order conditions for a maximum are satisfied

Answer 1

$(A)\\\pi=P1\times Q1+P2\times Q2-C\\\pi=(32-Q1)\times Q1+(40+2Q2)\times Q2-4\times (Q1+Q2)$

$\pi=32\times Q1-Q12+40\times Q2-2\times Q22-4\times Q1-4 \times Q2\\$

$\pi=28\times Q1-Q12+36\times Q2-2\times Q22$

For market 1 we obtain

$R1\times Q1=p1\times q1=(32-q1)\times q1\\=32\times q1-q1^2$

and hence 

$M\times R1=R1'\times q1=32-q1$

For market 2 we obtain

$R2\times Q2=p2\times q2=(40-2\times q2)\times q2=40\times q2-2\times q2^2$

and hence 

$M\times R2=R2'\times q2=40-2\times q2$

The monopolist will set marginal revenue in each market equal to the (common) marginal cost. Hence, in equilibrium, 

$M\times R1=32-q1=2(q1+q2)=M C\\M\times R2=40-2\times Q2=2(q1+q2)=MC$

This is an equation system with two equations and two unknown. From the first equation we obtain

$32-q1=2(q1+q2)\\32-q1=2\times q1+ 2\times q2$

which, solving for q1 in terms of q2, yields

$32=3\times q1+2\times q2\\3\times q1=32-2\times q2$

$q1=\frac{32-2\times q2}{3}$

Using this to replace q1 in the second equation then yields the following equation in q2

$40-2\times q2=2(\frac{32-2\times q2}{3}+q2)$

$40-2\times q2=2(\frac{32-2\times q2+3q2}{3})$

$40-2\times q2=\frac{64-4\times q2+6\times q2}{3}$

$120-6\times q2=64+2\times q2\\120-64=8\times q2\\56=8\times q2\\q2=7$

Using this equilibrium value to replace q2 in the equation for market 1 we then obtain

$q1=\frac{32-2\times 7}{3}=\frac{32-14}{3}=\frac{18}{3}=6$

$q1=6$

Hence, the quantities sold by the monopolist will be

$q1=6\\q2=7$